(untitled) (Statistics, Optimization & Information Computing, 2025)
‎In practice‎, ‎we may encounter hypotheses that the parameters under test have typical restrictions‎. ‎These restrictions can be placed in the null or alternative hypotheses‎. ‎In such a case‎, ‎the hypothesis is not included in the classical hypothesis testing framework‎. ‎Therefore‎, ‎statisticians are looking for the more powerful tests‎, ‎rather than the most powerful tests‎. ‎A common method for such tests is to use intersection-union and union-intersection tests‎. ‎In this paper‎, ‎we derived the testing procedure of a simple intersection-union and compared it with the likelihood ratio test‎. ‎We also compare the powers of two exponential sign tests‎, ‎the rectangle test and smoother test‎, ‎and the simple intersection-union test with the likelihood ratio test‎.